Definition 39.11.1. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a scheme over $S$, and let $a : G \times _ S X \to X$ be an action of $G$ on $X$.
We say $X$ is a pseudo $G$-torsor or that $X$ is formally principally homogeneous under $G$ if the induced morphism of schemes $G \times _ S X \to X \times _ S X$, $(g, x) \mapsto (a(g, x), x)$ is an isomorphism of schemes over $S$.
A pseudo $G$-torsor $X$ is called trivial if there exists an $G$-equivariant isomorphism $G \to X$ over $S$ where $G$ acts on $G$ by left multiplication.